Then, we prove that Protocol 3 can indeed output the LCM \(y= \underset{k\in \left[ n \right]}{\mathrm {lcm}}\left(x_k\right) \) . First, the correctness of the key QPA process (Algorithm 1) is proved[1]}, [2]}, [3]}, so we only need to prove that the period \(T\)  of the function \(f\)  is equal to \(y\) , as shown in Theorem  REF.
